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KINEMATIC ANALYSES OF A NEW MEDICAL ROBOT FOR3D VASCULAR ULTRASOUND EXAMINATION

Longfei Zhao, Andy Kar Wah Yen, Jonathan Coulombe, Pascal Bigras and Ilian A. BonevÉcole de Technologie Supérieure, Montreal, QC, Canada

E-mail: [email protected]

Received August 2013, Accepted March 2014No. 13-CSME-190, E.I.C. Accession 3648

ABSTRACTPeripheral artery disease (PAD) is a common vascular disease which can have serious consequences forolder people. Owing to the complexity of the vessels in the lower limbs, current PAD medical robots are notdesirable to diagnose PAD in this area. The kinematic model of a novel six-axis serial-parallel robot for 3Dvascular ultrasound examination of the lower limbs is presented in this paper. The prototype of the robot isdescribed, and then the direct and inverse kinematic problems are solved in closed form.

Keywords: serial-parallel robot; medical robot; kinematics.

MODÈLE GÉOMÉTRIQUE D’UN NOUVEAU ROBOT MÉDICALD’IMAGERIE 3D PAR ULTRASON

RÉSUMÉLa maladie occlusive artérielle périphérique affecte une partie importante de la population d’âge mûr. Lesrobots développés pour caractériser cette maladie ne sont actuellement pas adaptés aux membres inférieursen raison de la longueur et de la complexité du groupe d’artères concerné. Cet article présente le modèlegéométrique d’un nouveau robot sériel-parallèle conçu spécialement pour l’échographie tridimensionnelledes membres inférieurs. Le prototype est d’abord présenté. Puis, les modèles géométriques directe et inversesont expliqués en détail.

Mots-clés : robot série-parallèle; robot médical; modèle géométrique.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 2, 2014 227

1. INTRODUCTION

Peripheral arterial disease (PAD) is a very common, but serious disease that occurs in the lower limbs. Itnarrows, and can even block, the vessels that carry blood from the heart to the limbs [1]. According to [2],the prevalence of PAD of a group over 70 years old is nearly 15%, but only 10% of them have observablesymptoms. If PAD is not detected and diagnosed in time, patients will suffer intermittent claudication, andthe limb may even require amputation [3]. Medical imaging techniques are generally used in the diagnosis ofPAD to precisely locate blockages, or occlusions, and characterize their morphological features. Among themost common techniques, which include angiography, ultrasound scan, computed tomography angiography(CTA), and magnetic resonance angiography (MRA), ultrasound scan outweighs the others regarding to itsnon-radiation lower cost [4].

Several handheld medical robots have been developed for PAD diagnosis using ultrasound imaging, suchas TERESA [5, 6] and OTELO [7, 8]. This type of PAD robot requires quite a small workspace, and shouldbe held by the technician during operation. A survey of 232 sonographers [9] has revealed that the repetitivestrain of carrying this load over many working hours leads to musculoskeletal disorders. Several other robotshave been developed for ultrasound imaging of the abdominal area, such as Ehime University’s robot [10,11] and a cable robot, TER [12]. However, the abdominal area is flatter for ultrasound scan purposes thana leg, and the geometric design of these robots indicates that they are unsuitable for examining the lowerlimbs. The Hippocrate robot in France [13] and the fully statically balanced medical robot at the Universityof British Columbia [14] are designed for PAD diagnosis in the carotid area, where the artery concernedis fairly short and straight. Finally, in [15], we propose a new serial-parallel robot architecture for PADdiagnosis in the lower limbs, where blood vessels are long, complex, and twisted. That paper presents thearchitecture and a static balancing optimization for this robot, but its kinematic model is not analyzed, noris its prototype presented.

This paper presents hardware setup and a detailed kinematic analysis of the new medical robot (MedRUE)shown in Fig. 1(a), which is based on the architecture proposed in [15]. MedRUE (Medical Robot forvascular Ultrasound Examination) is suitable for lower limb PAD diagnosis as shown in Fig. 1(b), and theworkspace covers the entire lower limb area that needs to be scanned. Moreover, the robot has sufficientdexterity to insert and rotate a probe in the narrow space of a lower limb. The remaining sections areorganized as follows. In Section 2, the robot specifications and setup are discussed in detail. In Sections 3and 4, we present the direct and inverse kinematics of MedRUE respectively. In Section 5, we address thesingularity issues. Our conclusions are given in Section 6.

2. ROBOT ARCHITECTURE AND HARDWARE SETUP

MedRUE has a patented serial-parallel architecture [16] designed to follow the complex and twisted struc-ture of the artery in lower limbs. It can be regarded as comprising four components: a mobile base, twofive-bar mechanisms, and a tool part as shown in Fig. 1(b).

The mobile base is attached to the carriage of a LinTech 150 series linear guide. This allows MedRUEto translate along the x0 axis off the base frame O0, and it covers the length of a typical lower limb.This decoupled design has made it possible to minimize MedRUE’s dimensions. Furthermore, becausefour of its six motors are mounted on the mobile base, the rest links of the robot is relatively light andnimble.

The two symmetrical five-bar mechanisms are attached to the mobile base. They work in parallel planesperpendicular to the direction of the base linear guide. The combined motion of these two mechanismsenables the translation and orientation of the ultrasound probe along y0 and z0 axes.

The tool part connects the extremities of the two five-bar mechanisms through two passive universaljoints. A passive prismatic joint is located between the two universal joints to compensate for the change

228 Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 2, 2014

(a) (b)

Fig. 1. (a) MedRUE robot prototype; (b) MedRUE simulation.

Table 1. Specifications of the main components.Mass Length Other features

Robot 45 kg 0.91 m (linear guide) full extension: 0.687 mFive-bar mechanism inks

actuated bars 0.68 kg 0.40 m Inertia: 0.013 kg·m2

passive bars 0.82 kg 0.52 m Inertia: 0.024 kg·m2

Tool partTelescoping double universal joint 0.90 kg 0.080 m Radius: 0.030 mforce sensor and ultrasound probe 0.50 kg 0.176 m Radius: 0.020 m

in distance between the two extremities of the five-bar mechanisms. A small motor is located on one of themechanisms to drive the rotation of the tool part.

The general specifications of the robot and main components of MedRUE are listed in Table 1. They arebased on of an optimal design for MedRUE workspace as detailed in Section5. The main components aremade of aluminum alloy 6061-T6, and the shafts and pins are fabricated with alloy steel 4140.

The motors and drivers that were selected are listed in Table 2. The selection criterion mainly takes thesize, precision, and nominal speed into account. A Mini40 force/torque sensor from ATI is attached betweenthe ultrasound probe (only a dummy probe is shown in Fig. 1a) and the tool flange of the robot. Althoughit is small (a radius of 0.02 m and a height of 0.014 m), it can measure forces up to 60 N and torques upto 1 Nm. This is sufficient in ultrasound scan application, since the estimated force is typically below 10 Nand the torques are less than 1 Nm [17].

Figure 2 shows the setup of the robot system. A Q8 I/O card from Quanser provides eight A/D ports,eight D/A ports, eight encoder inputs, and thirty two digital I/O ports. Six A/D ports are used for reading theforce/torque sensor data, and six encoder inputs are used for position data. On a PC station, these sensor datacan be read and serve as feedback for the robot controller using the Quanser library QuaRC in Simulink.A real-time thread is created by QuaRC to run the program code generated by Simulink. While MedRUEis moving, the user can start/stop the program or modify/observe the controller’s parameters in Simulink.The Q8 card sends the control command from six D/A ports to drivers to generate torques for motors on therobot.


Table 2. MedRUE motors and drivers.Actuators Q1 Q2,Q3,Q4,Q5 Q6Drivers Danaher S20660VTS Danaher S20260VTS Maxon ADS 50/5Servomoters Kollmorgen AKM42G Kollmorgen AKM31E Maxon 31007SP

Mass 3.4 kg 1.6 kg 0.24 kgInertia 1.5×10−4 kg·m2 0.33×10−4 kg·m2 0.033×10−4 kg·m2

Gear box LinTech 150836-WC1-1-S129-M04 HD CSF-20-80 HD CSF-08-50Gear ratio 100 π rad: 1 m 80:1 50:1

Fig. 2. Setup of MedRUE.

3. DIRECT KINEMATIC MODEL

A geometric approach based on MedRUE’s four components is used to find its direct kinematic model.First, a kinematic model of a general five-bar mechanism is analyzed, and its results are then adapted toMedRUE’s kinematic model.

Figure 3 shows a general five-bar mechanism in the xiyi plane of the frame Oi (i = 1,2 throughout thispaper to indicate the number of five-bar mechanism) in two assembly modes. The link AiCi with length 2d1is fixed with the xi axis of frame Oi, while other four links can rotate with axes located at their geometricends. Lengths of four movable links are denoted as li j where j = 1 to 4. Variables qAi and qCi are the valuesof the actuated revolute joints, while qBi and qDi are passive revolute joints values. Ei is considered as theend point of this mechanism.

The vectors from the origin point Oi to Ai and Ci are represented as

rOiAi = [xOiAi yOiAi zOiAi ]T = [d1 0 0]T , (1)

rOiCi = [xOiCi yOiCi zOiCi ]T = [−d1 0 0]T . (2)

Thus the vectors from the origin point Oi to Bi and Di are obtained as

rOiBi(qAi) =

xOiAi + li1 cosqAi

yOiAi + li1 sinqAi

0

, (3)


(a) (b)

Fig. 3. General five-bar mechanism: (a) positive assembly mode; (b) negative assembly mode.

rOiDi(qCi) =

xOiCi + li3 cosqCi

yOiCi + li3 sinqCi

0

. (4)

As shown in Fig. 3, Si is defined as the projection of Ei on the vector rDiBi = rOiBi − rOiDi . If ||rDiBi || =√rT

DiBirDiBi > li4 + li2, there will be no solution, since the distance between qAi and qCi exceeds the sum of

the link lengths. Otherwise, applying the Pythagorean theorem on the two right triangles DiSiEi and BiSiEi,we have:

‖rSiEi‖2 +‖rDiSi‖2 = l2i4

‖rSiEi‖2 +(‖rDiBi‖−‖rDiSi‖)2 = l2i2

}. (5)

Equation (5) gives the solution:

rDiSi(qAi ,qCi) =l2i4− l2

i2 +‖rDiBi‖2

2‖rDiBi‖rDiBi

‖rDiBi‖

rSiEi(qAi ,qCi) =√

l2i4−‖rDiSi‖2

0 −1 01 0 00 0 0

rDiBi

‖rDiBi‖

. (6)

The solution for the end point Ei is

rOiEi(qAi ,qCi) = rOiDi + rDiSi± rSiEi . (7)

The sign in front of vector rSiEi is positive if the mechanism is in positive assembly mode, as indicated inFig. 3(a), and negative when in negative assembly mode, as shown in Fig. 3(b). The five-bar mechanism inMedRUE is the first case.

Then, the angle values of the passive joints are

qBi(qAi) = atan2(yOiEi− yOiBi ,xOiEi− xOiBi)−qAi , (8)


Fig. 4. MedRUE architecture.

qDi(qCi) = atan2(yOiEi− yOiDi ,xOiEi− xOiDi)−qCi . (9)

If the five-bar mechanism in Fig. 4 is rotated around the zi axis at a constant angle of −(θ +π/2), andthen duplicated onto the mobile base, the architecture of MedRUE in Fig. 1(b) can be redrawn as in Fig. 4.It is notable that both qBi and qDi in Fig. 4 are always greater than π rad, for providing space for patientsunder the robot arms. The two five-bar planes are parallel and perpendicular to the x0 axis of the base frameO0. Referring to the definition of q4 in Fig. 4, four actuated revolute joint values are defined as

q4 = π−θ +qA2 ,q5 = π−θ +qC2 ,q2 = π−θ +qA1 ,q3 = π−θ +qC1 . (10)

The homogeneous transformation matrix of the frames Oi w.r.t. the base frame O0 is

0Ti(q1) = Dx(xO0Oi)Dy(yO0Oi)Dz(zO0Oi)Ry

(π

2

)Rz(−θ) =

0 0 1 xO0Oi

−sin(θ) cos(θ) 0 yO0Oi

−cos(θ) −sin(θ) 0 zO0Oi

0 0 0 1

, (11)

where xO0Oi(q1) = d2 +q1 +(−1)id3, θ is a constant value in the mechanical design of MedRUE in Fig. 4.The coordinates of Ei can be represented in the base frame by parameters defined in Eq. (10), combined

with the five-bar model Eq. (7) and transformation Eq. (11)

rO0E1(q1,q2,q3) =0T1(q1)rO1E1(qA1 ,qC1), (12)

rO0E2(q1,q4,q5) =0T2(q1)rO2E2(qA2 ,qC2). (13)


(a) (b)

(c)

Fig. 5. Architecture of the MedRUE tool part: (a) overall structure; (b) microscopic view of the dashed block in (a);(c) view on (b) from right to left.

As shown in Fig. 5(a), the coordinates of the universal joint centers are

rO2F2(q1,q4,q5) = rO2E2− [d4 0 0]T , (14)

rO2F2(q1,q4,q5) = rO0E2− [d4 0 0]T . (15)

The dashed box in Fig. 5(b) demonstrates the physical revolution sequence of the tool part: rotation alongthe temporal axes x′, y′, and z′. Owing to the mechanical design of the tool part, x′ always aligns with x0,and z′ aligns with zP. As defined in Fig. 5(b), when α = β = γ = 0, then x′, y′ and z′ are parallel to thecorresponding axes of the base frame.

Confirming the revolution sequence of the tool part, Euler–XYZ angles [18] are chosen to express theorientation of the probe frame OP w.r.t. the base frame O0. In this context, γ is the sum of the rotationsalong x′ as in Fig. 5(c), and can be obtained by

γ(q2,q3,q6) = qD1E1 +q6, (16)

whereqD1E1(q2,q3) = atan2(−(yO0E1− yO0D1),zO0E1− zO0D1).


Actuator Q6 is fixed on l14 between D1 and E1, and so q6 is defined as the angle starting from l14 to thex′z′ plane. Since the probe will always point down to the skin surface during the scan process, γ is restrainedin the open interval (π/2,3π/2).

The unit vector from F1 to F2 is parallel to xP axis of the frame OP, yielding

uF1F2 = Rx(γ)Ry(β )Rz(α)[1 0 0]T =

cosβ cosα

sinγ sinβ cosα + cosγ sinα

−cosγ sinβ cosα + sinγ sinα

. (17)

Equation (17) can also be represented by variables defined in Eqs. (14) and (15)

uF1F2(q2,q3,q4,q5) =rO0F2− rO0F1

‖rO0F2− rO0F1‖=

ux

uy

uz

. (18)

If it were assumed that |α|, |β | can reach π/2, then the projection of F1 and F2 on x0 axis would be equalin Fig. 5. This is impossible, because F1 and F2 are rigidly attached to the two parallel five-bar mechanisms,and the distance along between F1 and F2 along x0 axis is a constant

ux‖rO0F2− rO0F1‖= xO0F2− xO)F1 = 2(d3−d4). (19)

Thus, the inequalities |α| < π/2 and |β | < π/2 must stand, and cosα 6= 0 in Eq. (17). In the design ofMedRUE, the mechanical limit of a universal joint is |α ≤ π/6, |β | ≤ π/6. Then, α and β can be computedby Eq. (17) and Eq. (18)

α(q2,q3,q4,q5) = sin−1(uy cosγ +uz sinγ), (20)

β (q2,q3,q4,q5) = sin−1(

uy sinγ−uz cosγ

cosα

). (21)

The coordinates of the probe tip P can be represented by

rO0P(q1,q2,q3,q4,q5,q6) = rO0F1 +0RPrF1P, (22)

where rF1P = [xF1P,0,zF1P]T is a constant vector in Fig. 5(a), 0RP(q2,q3,q4,q5,q6) = Rx(γ)Ry(β )Rz(α).

Thus the homogeneous transformation matrix of OP w.r.t. O0 is

0TP(q1,q2,q3,q4,q5,q6) =

[ 0RP rO0P

0 1

].

4. INVERSE KINEMATIC MODEL

The inverse kinematic model will be solved based on the MedRUE components, in the same way that thedirect kinematic model was solved in the previous section. In this case, the coordinates of Ei are com-puted based on the features of the tool part, and then be transformed into the local frames Oi of five-barmechanisms. Finally, all the joint values are obtained by solving the inverse kinematic model for a five-barmechanism.

Given the pose (xOoP,yO0P,zO0P,α,β ,γ) of the probe frame OP, the coordinates of F1 can be obtained byinverting Eq. (22):

rO0F1(xOoP,yO0P,zO0P,α,β ,γ) = rO0P(xOoP,yO0P,zO0P)−Rx(γ)Ry(β )Rz(α)rF1P. (23)


(a) (b)

Fig. 6. Two solutions for the inverse kinematic model of a five-bar mechanism.

To obtain the coordinates of F2, Eq. (18) can be rewritten as

rO0F2‖rO0F2− rO0F1‖uF1F2 + rO0F1 (24)

where uF1F2 can be calculated from Eq. (17). By combining Eqs. (17) and (19), the distance between F1 andF2 can be represented as

‖rO0F2− rO0F1‖=2(d3−d4)

cosβ cosα. (25)

Adding the offsets according to Eqs. (14) and (15), the end points of the two five-bar mechanisms are

rO0Ei = rO0Fi +(−1)i[d4 0 0]T , i = 1,2. (26)

From Eq. (11), the translation joint value is

q1 = xO0O1 +d3−d2. (27)

Since the five-bar mechanism is located in the plane perpendicular to the x0 axis of frame O0, Eq. (27)can be rewritten as

q1 = xO0E1 +d3−d2. (28)

With the constant transformation matrix 0Ti(θ) that we introduced in Eq. (11), the coordinates of the endpoints of the five-bar mechanisms can be represented in their local frame Oi, as in Eq. (29). In this way,the rest of the inverse geometric model of the robot is transformed into an inverse geometric model for afive-bar mechanism:

rOiEi =0T−1

i (q1)rO0Ei , for i = 1,2. (29)


4.1. Five-Bar MechanismThere will be no solution for qAi , if ‖rAiEi‖> li1 + li2, where rAiEi = rOiAi − rOiEi . As shown in Fig. 6, Gi isdefined as the projection of Bi on the vector rAiEi = rOiEi− rOiAi and Hi is defined as the projection of Di onthe vector rCiEi = rOiEi − rOiCi . Applying the same method used in Eq. (5) on both the left and right-handsides of the five-bar mechanism, we obtain

rAiGi =l2i1− l2

i2 +‖rAiEi‖2

2‖rAiEi‖rAiEi

‖rAiEi‖

rGiBi =√

l2i1−‖rAiGi‖2

0 −1 01 0 00 0 0

rAiEi

‖rAiEi‖

, (30)

rCiHi =l2i3− l2

i4 +‖rCiEi‖2

2‖rCiEi‖rCiEi

‖rCiEi‖

rHiDi =√

l2i3−‖rCiHi‖2

0 −1 01 0 00 0 0

rCiEi

‖rCiEi‖

. (31)

Confirming the MedRUE configuration, the vector rOiBi can be obtained by

rOiBi = rOiAi + rAiGi± rGiBi , (32)

rOiDi = rOiCi + rCiHi± rHiDi . (33)

The signs in Eqs. (32) and (33) demonstrate two different solutions of the configurations of a unit, giventhe pose of the end points of the five-bar mechanism. The sign is negative when the configuration is as inFig. 6(a), and positive when it is as shown in Fig. 6(b). MedRUE always works in the latter case for bothfive-bar mechanisms. Knowing the coordinates of Bi and Di, the active joint values can be obtained by

qAi = atan2(yOiBi− yOiAi ,xOiBi− xOiAi), (34)

qCi = atan2(yOiDi− yOiCi ,xOiDi− xOiCi). (35)

Finally, by submitting the results qAi and qCi to Eq. (10), we obtain the joint values q2, q3, q4, and q5. Thetranslation joint q1 is obtained in Eq. (28), and the last joint value can be deduced from Eqs. (9) and (16):

q6 = γ−q3−qD1 . (36)

5. SINGULARITY ISSUES AND WORKSPACE

Singularities only exist in the two five-bar mechanisms of MedRUE. For each mechanism, there are twotypes of singularity. A Type I singularity occurs when any side of the five-bar mechanism fully extends inFig. 7(a) or overlaps in Fig. 7(b). A Type II singularity occurs when two adjacent bars around the end pointoverlap in Fig. 7(c) or are aligned in Fig. 7(d). Because of the physical joint limits for qAi and qCi , only thesingularity in Fig. 7(a) is achievable in reality. A safety mechanism will freeze the motors when the robotapproaches the singularity region.

A simulation is done to demonstrate the workspace of MedRUE. Since linear motion of q1 dominates themotion along the x direction of base frame, the workspace of MedRUE can be obtained by extending itsworkspace in the y0z0 plane along the x0 axis. The position of probe in the y0z0 plane is mainly determined


(a) (b)

(c) (d)

Fig. 7. Singularities in the five-bar mechanism.

by the symmetrically assembled two five-bar mechanisms, thus, the analysis of the workspace of one five-barmechanism will lead to the workspace of MedRUE.

In Fig. 8, we take 22 samples for q2 and 17 samples for q3 throughout their joint limits respectively (qAi ∈[130◦,235◦], qCi ∈ [120◦,210◦]). After eliminating the counterpart solution as in Fig. 6(a), 298 positions ofEi are presented in small dotted circles in Fig. 8. The outlines of Ei positions are constituted by five curves,either due to joint limits or singularity case when qBi = 2π .

During the ultrasound scan process, the trajectory of probe will follow curves on the surface of the patientlegs. For simplification purposes, it can be approximated as a semicircle. In Fig. 8, the outer semicirclecurve (Ei-semicircle) is the maximum radius of semicircle the point Ei can reach in its workspace. With anoffset of the probe length, the inner semicircle curve (probe-semicircle) demonstrates the maximum radiusof semicircle for the probe when probe is pointing to the normal of the surface.

The physical orientation limit of MedRUE probe is [0◦ 360◦) along the x0 axis and [−30◦ 30◦] alongthe y0/z0 axes. In the scan process, only (0◦ 180◦) is needed for rotation along the x0 axis. Becauseof the offset of the probe length and the outer boundary Ei-semicircle in Fig. 8, only the points inside theprobe-semicircle can meet the orientation requirements for ultrasound scan process. The points outsideprobe-semicircle have difficulties to point to the circle center without violating the Ei-semicircle constrains.Rotations along the y0/z0 axis skew the probe tip off the y0z0 plane and it reduces the scan workspace alongthe x0 axis. To allow the robot to performance the orientation of y0/z0 axes, a distance of zF1P sin(π/6) mustbe reserved for the linear guide on each side of its limit. In summary, the workspace during ultrasound scan


Fig. 8. Workspace analysis of MedRUE.

is an extension of probe-semicircle along the x0 axis. Under the specification of Table 1, it is a semi-cylindershaped volume with 0.734 meter in length and 0.3 meter in diameter, (0◦ 180◦) along the x0 axis and[−30◦ 30◦] along the y0/z0 axes orientation.

6. CONCLUSIONS

A new medical robot, MedRUE, is presented in this paper. It has the ability to diagnose PAD in the large,complex and twisted arterial system of the lower limbs. An intuitive solution for its direct and inversekinematic model is discussed. Using this method, a complex serial-parallel robot system is decomposedinto several simplified sub-mechanisms. This system will be proposed to relieve sonographers of their dailyphysical load of carrying an ultrasound imaging equipment. The robot will not only help in the automaticdiagnosis of PAD, but also provide reliable data for 3D reconstruction in future research.

ACKNOWLEDGMENT

The authors thank the Fonds québécois de la recherche sur la nature et les technologies for providing thefunding for this project.

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